Nonlinear aspects of Calder on-Zygmund theory Giuseppe Mingione - - PowerPoint PPT Presentation

Nonlinear aspects of Calder´

• n-Zygmund theory

Giuseppe Mingione Ancona, June 7 2011

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Overture: The standard CZ theory

Consider the model case △u = f in Rn

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Overture: The standard CZ theory

Consider the model case △u = f in Rn Then f ∈ Lq implies D2u ∈ Lq 1 < q < ∞ with natural failure in the borderline cases q = 1, ∞

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Overture: The standard CZ theory

Consider the model case △u = f in Rn Then f ∈ Lq implies D2u ∈ Lq 1 < q < ∞ with natural failure in the borderline cases q = 1, ∞ As a consequence (Sobolev embedding) Du ∈ L

nq n−q

q < n

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The singular integral approach

Representation via Green’s function u(x) ≈

• G(x, y)f (y) dy

with G(x, y) =    |x − y|2−n if n > 2 − log |x − y| if n = 2 Differentiation yields D2u(x) =

• K(x, y)f (y) dy

and K(x, y) is a singular integral kernel, and the conclusion follows

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Singular kernels with cancellations

Initial boundedness assumption ˆ KL∞ ≤ B , where ˆ K denotes the Fourier transform of K(·) H¨

• rmander cancelation condition
• |x|≥2|y|

|K(x − y) − K(x)| dx ≤ B for every y ∈ Rn

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Another linear case

Higher order right hand side △u = div Du = div F Then F ∈ Lq = ⇒ Du ∈ Lq q > 1 just “simplify” the divergence operator!!

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Gradient integrability theory

Part 1: Gradient integrability theory

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The fundamentals

Theorem (Iwaniec, Studia Math. 83) div (|Du|p−2Du) = div (|F|p−2F) in Rn Then it holds that F ∈ Lq = ⇒ Du ∈ Lq p ≤ q < ∞

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The fundamentals

Theorem (Iwaniec, Studia Math. 83) div (|Du|p−2Du) = div (|F|p−2F) in Rn Then it holds that F ∈ Lq = ⇒ Du ∈ Lq p ≤ q < ∞ The local estimate

• −
• BR

|Du|q dz 1

q

≤ c

• −
• B2R

|Du|p dz 1

p

+ c

• −
• B2R

|F|q dz 1

q Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

General elliptic problems

In the same way the non-linear result of Iwaniec extends to all elliptic equations in divergence form of the type div a(Du) = div (|F|p−2F) where a(·) is p-monotone in the sense of the previous slides and to all systems with special structure div (g(|Du|)Du) = div (|F|p−2F)

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

General systems - the elliptic case

The previous result cannot hold for general systems div a(Du) = div (|F|p−2F) with a(·) being a general p-monotone in the sense of the previous slide. The failure of the result, which happens already in the case p = 2, can be seen as follows Consider the homogeneous case div a(Du) = 0 The validity of the result would imply Du ∈ Lq for every q < ∞, and, ultimately, that u ∈ L∞ But Sver´ ak & Yan (Proc. Natl. Acad. Sci. USA 02) recently proved the existence of unbounded solutions, even when a(·) is non-degenerate and smooth

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The up-to-a-certain-extent CZ theory

For general elliptic systems it holds Theorem (Kristensen & Min., ARMA 06) div a(Du) = div (|F|p−2F) in Ω for a(Du) being a p-monotone vector field and p ≤ q < p + 2p n − 2 + δ n > 2 Then it holds that F ∈ Lq

loc =

⇒ Du ∈ Lq

loc

Applications to singular sets estimates follow

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The up-to-a-certain-extent CZ theory

For general elliptic systems it holds Theorem (Kristensen & Min., ARMA 06) div a(Du) = div (|F|p−2F) in Ω for a(Du) being a p-monotone vector field and p ≤ q < p + 2p n − 2 + δ n > 2 Then it holds that F ∈ Lq

loc =

⇒ Du ∈ Lq

loc

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The Dirichlet problem

Theorem (Kristensen & Min., ARMA 06) Let −div a(Du) = 0 in Ω u = v

• n ∂Ω

with Ω being suitably regular (say C 1,α). Moreover, let p ≤ q < p + 2p n − 2 + δ n > 2 . Then it holds that

• Ω

|Du|q dx ≤ c

• Ω

(|Dv|q + 1) dx There is some evidence that the assumed bound on q is sharp

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The parabolic case

Theorem (Acerbi & Min., Duke Math. J. 07) ut − div (|Du|p−2Du) = div (|F|p−2F) in Ω × (0, T) for p > 2n n + 2 Then it holds that F ∈ Lq

loc =

⇒ Du ∈ Lq

loc

for p ≤ q < ∞

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The parabolic case

Theorem (Acerbi & Min., Duke Math. J. 07) ut − div (|Du|p−2Du) = div (|F|p−2F) in Ω × (0, T) for p > 2n n + 2 Then it holds that F ∈ Lq

loc =

⇒ Du ∈ Lq

loc

for p ≤ q < ∞ The lower bound p > 2n n + 2 is optimal

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The parabolic case

The elliptic approach via maximal operators only works in the case p = 2 The result also works for systems, that is when u(x, t) ∈ RN, N ≥ 1 First Harmonic Analysis free approach to non-linear Calder´

• n-Zygmund estimates

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The parabolic case

The result is new already in the case of equations i.e. N = 1, the difficulty being in the lack of homogenous scaling of parabolic problems with p = 2, and not being caused by the degeneracy of the problem, but rather by the polynomial growth. The result extends to all parabolic equations of the type ut − div a(Du) = div (|F|p−2F) with a(·) being a monotone operator with p-growth. More precisely we assume      ν(s2 + |z1|2 + |z2|2)

p−2 2 |z2 − z1|2 ≤ a(z2) − a(z1), z2 − z1

|a(z)| ≤ L(s2 + |z|2)

p−1 2

,

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The parabolic case

The result also holds for systems with a special structure (sometimes called Uhlenbeck structure). This means ut − div a(Du) = div (|F|p−2F) with a(·) being p-monotone in the sense of the previous slide, and satisfying the structure assumption a(Du) = g(|Du|)Du The p-Laplacean system is an instance of such a structure

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Pointwise estimates

Part 2: Pointwise estimates via nonlinear potentials

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The standard CZ theory

Consider the model case −△u = µ in Rn Then, if we define Iβ(µ)(x) :=

• Rn

dµ(y) |x − y|n−β , β ∈ (0, n] we have |u(x)| ≤ cI2(|µ|)(x) , and |Du(x)| ≤ cI1(|µ|)(x)

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Local versions

In bounded domains one uses Iµ

β(x, R) :=

R µ(B(x, ̺)) ̺n−β d̺ ̺ β ∈ (0, n] since Iµ

β(x, R)

• BR(x)

dµ(y) |x − y|n−β = Iβ(µB(x, R))(x) ≤ Iβ(µ)(x) for non-negative measures

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

What happens in the nonlinear case?

For instance for nonlinear equations with linear growth −div a(Du) = µ that is equations well posed in W 1,2 (p-growth and p = 2) And degenerate ones like −div (|Du|p−2Du) = µ

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The setting

We consider equations −div a(Du) = µ under the assumptions

• |a(z)| + |az(z)|(|z|2 + s2)

1 2 ≤ L(|z|2 + s2) p−1 2

ν−1(|z|2 + s2)

p−2 2 |λ|2 ≤ az(x, z)λ, λ

with p ≥ 2 this last bound is assumed in order to keep the exposition brief

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Non-linear potentials

The nonlinear Wolff potential is defined by Wµ

β,p(x, R) :=

R |µ|(B(x, ̺)) ̺n−βp

• 1

p−1 d̺

̺ β ∈ (0, n/p] which for p = 2 reduces to the usual Riesz potential Iµ

β(x, R) :=

R µ(B(x, ̺)) ̺n−β d̺ ̺ β ∈ (0, n] The nonlinear Wolff potential plays in nonlinear potential theory the same role the Riesz potential plays in the linear one

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

A fundamental estimate

For solutions to div (|Du|p−2Du) = µ with p ≤ n we have

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

A fundamental estimate

For solutions to div (|Du|p−2Du) = µ with p ≤ n we have Theorem (Kilpel¨ ainen-Mal´ y, Acta Math. 94) |u(x)| ≤ cWµ

1,p(x, R) + c

• −
• B(x,R)

|u|p−1 dy

• 1

p−1 Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

A fundamental estimate

For solutions to div (|Du|p−2Du) = µ with p ≤ n we have Theorem (Kilpel¨ ainen-Mal´ y, Acta Math. 94) |u(x)| ≤ cWµ

1,p(x, R) + c

• −
• B(x,R)

|u|p−1 dy

• 1

p−1

where Wµ

1,p(x, R) :=

R |µ|(B(x, ̺)) ̺n−p

• 1

p−1 d̺

̺ For p = 2 we have Wµ

1,p = Iµ 2

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

A fundamental estimate

For solutions to div (|Du|p−2Du) = µ with p ≤ n we have Theorem (Kilpel¨ ainen-Mal´ y, Acta Math. 94) |u(x)| ≤ cWµ

1,p(x, R) + c

• −
• B(x,R)

|u|p−1 dy

• 1

p−1

where Wµ

1,p(x, R) :=

R |µ|(B(x, ̺)) ̺n−p

• 1

p−1 d̺

̺ For p = 2 we have Wµ

1,p = Iµ 2

Another approach to this result has been given by Trudinger & Wang (Amer. J. Math. 02)

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Integral estimates follow via Wolff inequalities

We have µ ∈ Lq = ⇒ Wµ

β,p ∈ L

nq(p−1) n−qpβ

q ∈ (1, n) with related explicit estimates, also in Marcinkiewicz spaces Such a property allows to reduce the study of integrability of solutions to that of nonlinear potentials The key is the following inequality: ∞ |µ|(B(x, ̺)) ̺n−βp

• 1

p−1 d̺

̺ ≤ cIβ

• [Iβ(|µ|)]

1 p−1

• (x)

the last quantity is called Havin-Maz’ja potential

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The potential gradient estimate for p = 2

Theorem (Min., JEMS 11) |Dξu(x)| ≤ cI|µ|

1 (x, R) + c −

• B(x,R)

|Dξu| dy holds for almost every point x and ξ ∈ {1, . . . , n}

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The general potential gradient estimate

Theorem (Duzaar & Min., Amer. J. Math. 201?) |Du(x)| ≤ cWµ

1/p,p(x, R) + c −

• B(x,R)

(|Du| + s) dy holds for almost every point x

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The general potential gradient estimate

Theorem (Duzaar & Min., Amer. J. Math. 201?) |Du(x)| ≤ cWµ

1/p,p(x, R) + c −

• B(x,R)

(|Du| + s) dy holds for almost every point x This means |Du(x)| ≤ c R |µ|(B(x, ̺)) ̺n−1

• 1

p−1 d̺

̺ + c −

• B(x,R)

(|Du| + s) dx

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Integral estimates follow as a corollary

For the model case equation div (|Du|p−2Du) = µ the previous estimate implies for instance all those included in the papers Iwaniec, Studia Math. 83 Di Benedetto & Manfredi, Amer. J. Math. 93 Boccardo & Gall¨

• uet, J. Funct. Anal. 87, Comm PDE 92

Talenti, Ann. SNS Pisa 76 Kilpelainen & Li, Diff. Int. Equ. 00 Dolzmann & Hungerb¨ uhler & M¨ uller, Crelle J. 00 Alvino & Ferone & Trombetti, Ann. Mat. Pura Appl. 00 Boccardo, Ann. Mat. Pura Appl. 08 Moreover, the borderline cases which appeared as open problems in some of the above papers papers now follow as a corollary

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The two estimates

The two potential estimates are |u(x)| ≤ cWµ

1,p(x, R) + c −

• B(x,R)

(|u| + Rs) dy and |Du(x)| ≤ cWµ

1/p,p(x, R) + c −

• B(x,R)

(|Du| + s) dy They basically provide size estimates on u and Du The aim is now to provide estimates on the oscillations

• f solutions and/or alternatively, on intermediate

derivatives

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Calder´

• n spaces of DeVore & Sharpley

The following definition is due to DeVore & Sharpley (Mem. AMS, 1982) Let α ∈ (0, 1], q ≥ 1, and let Ω ⊂ Rn be a bounded open

• subset. A measurable function v, finite a.e. in Ω, belongs to

the Calde´ ron space C α

q (Ω) if and only if there exists a

nonnegative function m ∈ Lq(Ω) such that |v(x) − v(y)| ≤ [m(x) + m(y)]|x − y|α holds for almost every couple (x, y) ∈ Ω × Ω.

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Calder´

• n spaces of DeVore & Sharpley

In other words m(x) ≈ ∂αv(x) Indeed DeVore & Sharpley take Mα

#v(x) = sup B(x,̺)

̺−α −

• B(x,̺)

|v(y) − (v)B(x,̺)| dy For α ∈ (0, 1) and q > 1 we have W α,q ⊂ C α,q ⊂ W α−ε,q therefore such spaces, although not being of interpolation type, are just another way to say “fractional differentiabiltiy”

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Classical regularity review

For equations as div (γ(x)|Du|p−2Du) = 0 we have If γ(x) is measurable then u ∈ C 0,αm for some αm > 0 If γ(x) is VMO then u ∈ C 0,α for every α < 1 If γ(x) is Dini then u ∈ C 0,1 If γ(x) is H¨

• lder the Du ∈ C 0,αM for some αM > 0

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

General quasilinear equations

The previous and forthcoming results hold for general quasilinear equations div a(x, Du) = 0 − div a(x, Du) = µ where x → a(x, ·) is just measurable/VMO/Dini

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The homogeneous case

The classical estimate |u(x) − u(y)| ≤ −

• BR

(|u| + Rs) dξ · |x − y| R αm when coefficients are measurable

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The homogeneous case

The classical estimate |u(x) − u(y)| ≤ −

• BR

(|u| + Rs) dξ · |x − y| R α when coefficients are VMO-regular

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The homogeneous case

The classical estimate |u(x) − u(y)| ≤ −

• BR

(|u| + Rs) dξ · |x − y| R α holds for α ∈ (0, αm], (0, 1), (0, 1] depending on the regularity

• f coefficients

Moreover |Du(x) − Du(y)| ≤ −

• BR

(|Du| + s) dξ · |x − y| R α whenever α ≤ αM

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The first universal potential estimate

In the case of measurable coefficients we have Theorem (Kuusi & Min.) The estimate |u(x) − u(y)| ≤ c

• Wµ

1− α(p−1)

p

,p(x, R) + Wµ 1− α(p−1)

p

,p(y, R)

• |x − y|α

+c −

• BR

(|u| + Rs) dξ · |x − y| R α holds uniformly in every compact subset of α ∈ [0, αm), whenever x, y ∈ BR/4 The case α = 0 gives back the known potential estimate of Kilpel¨ ainen & Mal´ y as endpoint case

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The second universal potential estimate

In the case of VMO coefficients we have Theorem (Kuusi & Min.) The estimate |u(x) − u(y)| ≤ c

• Wµ

1− α(p−1)

p

,p(x, R) + Wµ 1− α(p−1)

p

,p(y, R)

• |x − y|α

+c −

• BR

(|u| + Rs) dξ · |x − y| R α holds uniformly in every compact subset of α ∈ [0, 1), whenever x, y ∈ BR/4

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The third universal potential estimate

Finally, in the case of Dini coefficients we have Theorem (Kuusi & Min.) The estimate |u(x) − u(y)| ≤ c

• Wµ

1− α(p−1)

p

,p(x, R) + Wµ 1− α(p−1)

p

,p(y, R)

• |x − y|α

+c −

• BR

(|u| + Rs) dξ · |x − y| R α holds uniformly in α ∈ [0, 1], whenever x, y ∈ BR/4 The cases α = 0 and α = 1 give back the two known potential estimates as endpoint cases

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The fourth universal potential estimate

In the case coefficients are H¨

• lder continuous we have

Theorem (Kuusi & Min.) The estimate |Du(x) − Du(y)| ≤ c

• Wµ

1− (1+α)(p−1)

p

,p(x, R) + Wµ 1− (1+α)(p−1)

p

,p(y, R)

• |x − y|α

+c −

• BR

(|Du| + s) dξ · |x − y| R α uniformly α belong to any compact subset of [0, αM), whenever x, y ∈ BR/4 The case α = 0 gives back the gradient potential estimate

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The universal potential estimate

With the previous terminology we have ∂αu ≤ cWµ

1− α(p−1)

p

,p

and ∂αDu ≤ cWµ

1− (1+α)(p−1)

p

,p

As a corollary we have optimal regularity criteria in oscillation spaces, such H¨

• lder spaces, Fractional spaces, and so on, both

for u and for the gradient Du

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

A fully fractional approach

Part 3: A fully fractional approach

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The setting for p = 2

We consider equations −div a(Du) = µ under the assumptions

• |a(z)| + |az(z)||z| ≤ L|z|

ν−1|λ|2 ≤ az(z)λ, λ

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

The setting for p = 2

Theorem (Min., JEMS 11) |Dξu(x)| ≤ cI|µ|

1 (x, R) + c −

• B(x,R)

|Dξu| dx for every ξ ∈ {1, . . . , n}, where Iµ

β(x, R) :=

R µ(B(x, ̺)) ̺n−1 d̺ ̺

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Classical Gradient estimates

Consider energy solutions to div a(Du) = 0 for p = 2 First prove Du ∈ W 1,2 Then use that v = Dξu solves div(A(x)Dv) = 0 A(x) := az(Du(x)) The boundedness of Dξu follows by Standard DeGiorgi’s theory This is a consequence of Caccioppoli’s inequalities of the type

• BR/2

|D(Dξu − k)+|2 dy ≤ c R2

• BR

|(Dξu − k)+|2 dy where (Dξu − k)+ := max{Dξu − k, 0}

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Recall the definition

We have v ∈ W σ,1(Ω′) iff v ∈ L1(Ω′) and [v]σ,1;Ω′ =

• Ω′
• Ω′

|v(x) − v(y)| |x − y|n+σ dx dy < ∞

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

There is a differentiability problem

For solutions to div a(Du) = µ in general Du ∈ W 1,1 but nevertheless it holds Theorem (Min., Ann. SNS Pisa 07) Du ∈ W 1−ε,1

loc

(Ω, Rn) for every ε ∈ (0, 1) This means that [Du]1−ε,1;Ω′ =

• Ω′
• Ω′

|Du(x) − Du(y)| |x − y|n+1−ε dx dy < ∞ holds for every ε ∈ (0, 1), and every subdomain Ω′ ⋐ Ω

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Step 1: A non-local Caccioppoli inequality

Theorem (Min., JEMS 11) Let w = Dξu with − div a(Du) = µ where ξ ∈ {1, . . . , n} then [(|w| − k)+]σ,1;BR/2 ≤ c Rσ

• BR

(|w| − k)+ dy + cR|µ|(BR) Rσ holds for every σ < 1/2

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Step 1: A non-local Caccioppoli inequality

Theorem (Min., JEMS 11) Let w = Dξu with − div a(Du) = µ where ξ ∈ {1, . . . , n} then [(|w| − k)+]σ,1;BR/2 ≤ c Rσ

• BR

(|w| − k)+ dy + cR|µ|(BR) Rσ holds for every σ < 1/2 Compare with the usual one for div a(Du) = 0, that is [(w − k)+]2

1,2;BR/2 ≡

• BR/2

|D(w − k)+|2 dy ≤ c R2

• BR

(w − k)2

+ dy

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Step 1: A non-local Caccioppoli inequality

This approach reveal the robustness of energy inequalities, which hold below the natural growth exponent 2, and for fractional order of differentiability, although the equation has integer order Classical VS fractional classical fractional spaces L2 − L2 L1 − L1 differentiability 0 − → 1 0 − → σ

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Step 2: Fractional De Giorgi’s iteration

Theorem (Min., JEMS 11) Let w be an L1-function w satisfying the fractional Caccioppoli’s inequality [(|w| − k)+]σ,1;BR/2 ≤ L Rσ

• BR

(|w| − k)+ dy + LR|µ|(BR) Rσ for some σ > 0 and every k ≥ 0. Then it holds that |w(x)| ≤ cI|µ|

1 (x, R) + c −

• B(x,R)

|w| dy for every Lebesgue point x of w

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory

Step 2: Fractional De Giorgi’s iteration

Theorem (Min., JEMS 11) Let w be an L1-function w satisfying the fractional Caccioppoli’s inequality [(|w| − k)+]σ,1;BR/2 ≤ L Rσ

• BR

(|w| − k)+ dy + LR|µ|(BR) Rσ for some σ > 0 and every k ≥ 0. Then it holds that |w(x)| ≤ cI|µ|

1 (x, R) + c −

• B(x,R)

|w| dy for every Lebesgue point x of w Proof of the gradient potential estimate: apply the previous result to w ≡ Dξu

Giuseppe Mingione Nonlinear aspects of Calder´

• n-Zygmund theory